What methods are there to find integers from given GCDs? I need to find the natural numbers a, b, c that satisfy gcd(a,b)=21, gcd(b,c)=21, and gcd(a,c)=819.
I managed to find a set {a=819,b=462,c=819} by some guessing and checking with Mathematica, but I need to be able to do this by hand. I had hoped to reverse engineer some method with a solution and the GCD properties, but so far I'm not having much success. Any help appreciated, thanks.
Let $\dfrac aA=\dfrac bB=21$ so that $(A,B)=1$
Similarly $b=21D,c=21C,(D,C)=1$
$\implies B=D,(B,C)=1$
Finally we need $819=(21A,21C)=21(A,C)\implies (A,C)=39$
WLOG $\dfrac Ap=\dfrac Cq=39,$ so that $(p,q)=1$
$(B,39p)=(B,39q)=(p,q)=1$
where $b=21B,a=21A=819p,c=819q$